1. In bacteriological investigations of stretches of water an estimate of the number of a certain type of bacterium in a unit of volume of the water is required so that the pattern of the pollution of the area can be discerned. Usually it is known or guessed that the density of bacteria,δ say, lies in a range δL up to δH. Several samples of water are taken and each is tested for a positive reaction which would indicate the presence of at least one bacterium of the type being searched for. When it is assumed that bacteria are spread in a Poisson manner through the water in each sample an estimate of their density can be made from the number of positively reacting samples.If the range δL to δH is large it is unlikely that one volume of water for each sample will give a very efficient estimate of the density;e.g. if this volume is in an average position with respect to (δL to δH), say, 2/(δL +δH), and the true density is near an end of the range, all the reactions are likely to have the same sign and the sample gives no discrimination. Even worse estimation for some densities will occur if a very small or very large volume is used. The setting of reasonably sized confidence intervals for the density requires a series of volumes designed to locate the density with about equal accuracy no matter where it is in (δL to δH). Hence the usual set of volumes is the dilution series discussed by several writers, for example recently by Cochran (1950).